Bring a saucepan of salted water to a boil. Fill a bowl with ice water. Cut off 1 inch of asparagus tips and cut each tip in half lengthwise. Using a peeler or mandoline, shave asparagus spears lengthwise. Blanch asparagus tips and strands in boiling water until just heated through, 10 seconds. Drain and transfer to ice water to chill; drain well. Pat dry and transfer to a bowl.

In a blender, blend oil, lemon juice, basil, shallot, and 1/4 cup parsley. Season dressing with salt and pepper.

Add pea shoots, pistachios, and the remaining 1/4 cup parsley leaves to asparagus. Toss with half dressing and season with salt and pepper. Add crab, avocado, and remaining dressing, toss gently, and serve at once.